Redundant core in multicore optical fiber for safety

ABSTRACT

An optical fiber includes multiple optical cores configured in the fiber including a set of primary cores and an auxiliary core. An interferometric measurement system uses measurements from the multiple primary cores to predict a response from the auxiliary core. The predicted auxiliary core response is compared with the actual auxiliary core response to determine if they differ by more than a predetermined amount, in which case the measurements from the multiple primary cores may be deemed unreliable.

This application is the U.S. national phase of International ApplicationNo. PCT/US2017/029568 filed Apr. 26, 2017, which designated the U.S. andclaims the priority and benefit of U.S. Provisional Patent Application62/334,649, filed May 11, 2016, entitled “REDUNDANT CORE IN MULTICOREOPTICAL FIBER FOR SAFETY,” the entire contents of each of which arehereby incorporated by reference.

BACKGROUND

Spun multicore fiber has been used to determine the shape of an opticalfiber. A multicore fiber having four cores can be used to separate thedeformation of the fiber into two bend angles (pitch and yaw), one twistangle, and the fiber elongation. These four measurements constitute fourdegrees of freedom. These four measurements (pitch, yaw, twist, andelongation) also represent all of the deformations that can happen tothe fiber with relatively small forces.

In fiber optic based shape sensing, a multi-channel distributed strainsensing system is used to detect the change in strain for each ofseveral cores within a multicore optical shape sensing fiber asdescribed in U.S. Pat. No. 8,773,650, incorporated herein by reference.Multiple distributed strain measurements are combined through a systemof equations to produce a set of physical measurements includingcurvature, twist, and axial strain as described in U.S. Pat. No.8,531,655, incorporated herein by reference. These physical measurementscan be used to determine the distributed shape and position of theoptical fiber.

Some applications for shape sensing fiber require a high degree ofconfidence or safety in terms of the accuracy and reliability of theshape sensing output. An example application is robotic arms used infine manufacturing, surgical, or other environments.

Another problem with shape sensing fiber applications is unforeseen orunpredictable errors that are not included in shape sensing models ormodel assumptions. Example errors include errors in the operation of theoptical and/or electronic sensing and processing circuitry, errors inconnecting fibers, human errors such as loading an incorrect calibrationfile to calibrate the shape sensing system, and errors caused by forcesexperienced by the fiber that are not included in the shape sensingmodel. One such parameter already described is fiber pinch. Anotherparameter is temperature if the shape sensing model does not account forchanges due to temperature. A further concern is other parameters notyet known or identifiable. So a further need is for the technologicalsolution to be able to detect errors that are independent from and notaccounted for in the shape sensing model.

SUMMARY

The technology in this application uses a model having N degrees offreedom and N measurements to predict an additional measurement. Inother words, the N degrees of freedom model is used to make N+1measurements, and the extra measurement made using an extra or redundantcore in the fiber is used as a check on the model. For example, with afive degree of freedom model of an optical shape sensing fiber havingsix optical cores, (the sixth core is an auxiliary or redundant core),where each segment of fiber can experience pitch, yaw, roll, tension,and spatial or temporal changes in temperature, five strains aremeasured and used to uniquely determine each of the five parameters(pitch, yaw, etc.) in the model. The determined five parameters are thenused to predict what the strain in the sixth core should be if the modelis correct and reliable, and the predicted strain is compared to themeasured strain in the sixth core to determine an error. Advantageously,the technology does not require advance knowledge of any specific errorin the model in order to detect that error and also detects errors ofunknown origin. Adding more auxiliary or redundant signals (7 coresinstead of 6 cores for the five degree of freedom model) furtherincreases confidence and trust in shape sensing measurements.

In example embodiments, an interferometric measurement system isprovided for measuring an optical fiber including multiple primary coresconfigured in the fiber and an auxiliary core configured in the fiber.Interferometric detection circuitry is configured to detect measurementinterferometric pattern data associated with each of the multipleprimary cores and the auxiliary core. This may be done when the opticalfiber is placed into a sensing position. Data processing circuitry isconfigured to determine compensation parameters based on the detectedmeasurement interferometric pattern data for the primary multiple cores,compare a predicted parameter value for the auxiliary core used and ameasurement-based parameter value for the auxiliary core to produce acomparison, determine an unreliability of the determined compensationparameters based on the comparison, and generate a signal in response tothe unreliability. The compensation parameters compensate for variationsbetween a calibration configuration of the multiple primary cores and anactual configuration of the multiple primary core.

The signal may represent an error comprising one or more of thefollowing: (a) an error in operation of the detection or data processingcircuitry, (b) an error in an optical fiber connection, (c) an error inthe calibration configuration, or (d) an error caused by a forceexperienced by the fiber for which a compensation parameter is notdetermined by the data processing circuitry. For example, theunreliability may be caused by a pinching of the optical fiber, causedby spatial or temporal changes in temperature, caused by some otherphenomenon that changes the measured signals from the optical fiber, orcaused by a combination thereof.

In one example application, the data processing circuitry is configuredto apply the compensation parameters to subsequently-obtainedmeasurement interferometric pattern data for the fiber.

In an example application, the predicted parameter value is a predictedphase for the auxiliary core and the measurement-based parameter valueis measurement-based phase value for the auxiliary core. The dataprocessing circuitry is configured to determine predicted phase for theauxiliary core by performing the following operations: calculate aderivative of a phase measured in each of the primary cores to obtainmultiple phase derivatives; multiply the multiple phase derivatives by aconversion matrix to obtain a predicted auxiliary core phase derivative;and integrate the predicted auxiliary core phase derivative to obtainthe predicted phase for the auxiliary core.

In another example application, the data processing circuitry isconfigured to: determine strain values for the fiber corresponding to anaxial strain, a bend strain, and a twist strain on the optical fiberbased on the detected measurement interferometric pattern data, anddetermine a shape of the optical fiber based on the determined strainvalues for the optical fiber corresponding to the axial strain, bendstrain, and twist strain on the optical fiber.

In another example application, the data processing circuitry isconfigured to: determine strain values for the fiber corresponding to anaxial strain, a bend strain, a twist strain, and a temperature strain onthe optical fiber based on the detected measurement interferometricpattern data, and determine a shape of the optical fiber based on thedetermined strain values for the optical fiber corresponding to theaxial strain, bend strain, twist strain, and temperature strain on theoptical fiber. Temperature strain is used herein to indicate straincaused by temperature, such as caused by spatial or temporal changes intemperature occurring after calibration or after reference baselinereadings are taken.

In an example implementation, the data processing circuitry isconfigured to generate the signal when the unreliability exceeds apredetermined threshold.

In another example implementation, the signal is representative of theunreliability.

Other example embodiments include an interferometric measurement methodfor measuring an optical fiber including multiple primary coresconfigured in the fiber and an auxiliary core configured in the fiber.The method includes:

detecting, using interferometric detection circuitry, measurementinterferometric pattern data associated with each of the multipleprimary cores and the auxiliary core when the optical fiber is in asensing position; and

determining, using data processing circuitry, compensation parametersbased on the detected measurement interferometric pattern data for themultiple primary cores, the compensation parameters compensating forvariations between a calibration configuration of the multiple primarycores and an actual configuration of the multiple primary cores,

comparing a predicted parameter value for the auxiliary core with ameasurement-based parameter value for the auxiliary core to produce acomparison,

determining an unreliability of the compensation parameters based on thecomparison, and

generating a signal indicating in response to the unreliability.

Other example embodiments include an optical fiber with five or morecores including a central core and four or more peripheral cores each ata radius distance from the central core. One or more of the five or morecores provides a temperature response different than a temperatureresponse of the other cores. One of more of the four or more peripheralcores is at a first radius distance from the central core different froma second radius distance from the central core associated with the otherfour or more peripheral cores. The difference between the first radiusdistance and the second radius distance is at least 10% of an averageradius distance associated with all of the five or more cores.

In an example implementation, the five or more cores arehelically-twisted along a length of the optical fiber.

In another example implementation, the one or more cores providing adifferent temperature response has/have a doping or material differentthan the other cores.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows a robotic arm with a rotatable joint that illustrates anexample of fiber pinch in the joint.

FIG. 2 shows a first twisted multicore fiber example embodiment withfive cores.

FIG. 3 illustrates mathematical parameters that can be used to quantifycore placement and a response to strain for a five core, helicallytwisted optical fiber.

FIG. 4 shows a schematic diagram of a first example embodiment of anoptical frequency domain reflectometry (OFDR)-based shape sensing systemthat identifies errors independent from the model and assumptions of theshape sensing system using a five core fiber.

FIG. 5 is a flowchart diagram for calibrating the optical shape sensingfiber in the first example embodiment.

FIG. 6 is a flowchart diagram for detecting an error in shape sensingsystem of the first example embodiment.

FIG. 7 shows a graph of index of refraction versus temperature for aboron-germanium co-doped core and a germanium only doped core.

FIG. 8 shows a second twisted multicore fiber example embodiment withsix cores.

FIG. 9 shows a schematic diagram of a second example embodiment of anOFDR-based shape sensing system that compensates for temperature andidentifies errors independent from the model and assumptions of theshape sensing system using a six core fiber.

FIG. 10 is a flowchart diagram for calibrating the optical shape sensingfiber in the second example embodiment.

FIG. 11 is a flowchart diagram for detecting an error in shape sensingsystem of the second example embodiment.

DETAILED DESCRIPTION

The following description sets forth specific details, such asparticular embodiments for purposes of explanation and not limitation.But it will be appreciated by one skilled in the art that otherembodiments may be employed apart from these specific details. In someinstances, detailed descriptions of well-known methods, interfaces,circuits, and devices are omitted so as not to obscure the descriptionwith unnecessary detail. Individual blocks are shown in the figurescorresponding to various nodes. Those skilled in the art will appreciatethat the functions of those blocks may be implemented using individualhardware circuits, using software programs and data in conjunction witha suitably programmed digital microprocessor or general purposecomputer, and/or using applications specific integrated circuitry(ASIC), and/or using one or more digital signal processors (DSPs).Software program instructions and data may be stored on anon-transitory, computer-readable storage medium, and when theinstructions are executed by a computer or other suitable processorcontrol, the computer or processor performs the functions associatedwith those instructions.

Thus, for example, it will be appreciated by those skilled in the artthat diagrams herein can represent conceptual views of illustrativecircuitry or other functional units. Similarly, it will be appreciatedthat any flow charts, state transition diagrams, pseudocode, and thelike represent various processes which may be substantially representedin computer-readable medium and so executed by a computer or processor,whether or not such computer or processor is explicitly shown.

The functions of the various illustrated elements may be providedthrough the use of hardware such as circuit hardware and/or hardwarecapable of executing software in the form of coded instructions storedon computer-readable medium. Thus, such functions and illustratedfunctional blocks are to be understood as being eitherhardware-implemented and/or computer-implemented, and thus,machine-implemented.

In terms of hardware implementation, the functional blocks may includeor encompass, without limitation, a digital signal processor (DSP)hardware, a reduced instruction set processor, hardware (e.g., digitalor analog) circuitry including but not limited to application specificintegrated circuit(s) (ASIC) and/or field programmable gate array(s)(FPGA(s)), and (where appropriate) state machines capable of performingsuch functions.

In terms of computer implementation, a computer is generally understoodto comprise one or more processors or one or more controllers, and theterms computer, processor, and controller may be employedinterchangeably. When provided by a computer, processor, or controller,the functions may be provided by a single dedicated computer orprocessor or controller, by a single shared computer or processor orcontroller, or by a plurality of individual computers or processors orcontrollers, some of which may be shared or distributed. Moreover, theterm “processor” or “controller” also refers to other hardware capableof performing such functions and/or executing software, such as theexample hardware recited above.

Example spun or helically-twisted multicore fibers are described belowfor purposes of illustration and not limitation. The principlesdescribed also apply to a multicore fiber where multiple primary coresand one or more secondary (e.g., redundant or auxiliary) cores havedifferent relative positions along a length of the optical fiber.

Because the outer cores of a spun fiber are helically-wrapped, the outercores also experience strain as a result of a twist applied to thefiber. The outer cores are either elongated or compressed in response tothe orientation of the twist to the direction of the helical wrapping.In other words, looking down the axis of the fiber with the outer coresbeing helically wrapped clockwise, an applied twist in the clockwisedirection causes the outer cores to become compressed. Conversely, acounter-clockwise applied twist causes the outer cores to elongate(experience tension). But the center core does not experience strain asa result of twist because it is placed along the neutral axis. Thus, afour-core fiber has sufficient degrees of freedom to allow individualdetermination of each of three different types of strain that can beapplied to the four-core fiber: axially-applied strain, bend-inducedstrain, and strain as a result of twist or torsion. The measured signalsfrom four cores are used to extract four linearly independent parametersthat describe a physical state of the fiber. Those four parametersinclude common mode strain, pitch bending, yaw bending, and twist, andthey represent relatively low force changes that can be imposed on thefiber.

Pinching of the fiber is another independent modification of the fiber.Inducing significant dimensional changes via pinching requiresrelatively large force as compared to the low forces above.

FIG. 1 shows a robotic arm 2 having a multi-core, shape sensing fiber 1.The robotic arm 2 includes a joint 3 that permits adjoining members ofthe robotic arm to rotate with respect to each other around the joint 3.There may be situations where, as the members of the robotic arm move,the fiber 1 may be pinched in the joint 3, as shown in the expanded view4. That pinching of the fiber introduces an error into the shape sensingdeterminations. In other words, pinching is an additional force notaccounted for in the four degree of freedom model that includes commonmode strain, pitch bending, yaw bending, and twist.

FIG. 2 shows a sensing fiber 1 that is a twisted multicore fiber withfive cores a-e. Core a is on or near the neutral axis, and cores b-e areintentionally offset from the neutral axis by a certain radius distance.As explained in conjunction with FIG. 3, the radial distances for allthe offset cores are not all the same.

FIG. 3 illustrates mathematical parameters that can be used to quantifycore placement and a response to strain for a five core, helicallytwisted optical fiber. Note that peripheral cores b-d are at the sameradius from the center of the multi-core fiber, and peripheral core e isat a different radius than the radii of peripheral cores b-d. In thisexample, the radius of an auxiliary core e is less than that of coresb-d, but the radius of core e may be more than that of cores b-d. Thedifference between the radius distances for cores b-d and the radiusdistance for core e is more than an insubstantial difference. Forexample, the difference divided by the average radius distance for allof cores b-e is 0.10 or more. Stated differently, the difference is 10%of the average radius distance for all of cores b-e. The auxiliary coree is preferably located in the fiber in such a way so as to reducecoupling with cores a-d. In some embodiments, the average radiusdistance is the mean radius distance.

In an alternative example embodiment, the auxiliary redundant core e islocated at the same radius as the other peripheral cores. Although theauxiliary redundant core may still be used to check the reliability ofthe data, this configuration is less effective at detecting errors onthe central core.

FIG. 3 also illustrates mathematical parameters that can be used toquantify core placement and a response to strain for a five core,helically twisted optical fiber. A vertical axis is placed through thecenter of the multi-core fiber such that it passes through one of theouter cores b. The outer core b that is bisected by the vertical axis isreferred to as the “reference core.” Note that several parameters willbe expressed relative to this core b and for the rest of this documentthe core identified with index n=1 serves as the reference core as anexample. Two parameters describe the position of a core: the radialdistance from the fiber center, r, and an arbitrary angle ϕ measuredfrom the established vertical axis intersecting the reference core. Asthe fiber is bent, the amount of bend-induced strain in a given core isdirectly proportional to the perpendicular distance d that a core isseparated from the bend plane. This is illustrated in the right diagramfor the outer core c index n=2. If the bend plane is described by theangle θ, the nature of the helical wrapping of the cores within thefiber may be determined. According to distance along the length of thefiber, θ is defined by the spin frequency of the helical fiber.

It is helpful to understand how these parameters impact the componentsof the strain profile of the fiber when the core strain responses arerecombined. A mathematical model is established based on the parametersshown in FIG. 3. Because these parameters can be measured, they can beused to provide a more accurate recombination of the strain profile ofthe multi-core optical fiber. It is notable that these parameters needonly be measured once for a particular multi-core optical fiber and maybe used for some or all OFDR subsequent measurements of that samemulti-core optical fiber.

As explained earlier, the strain applied to the multi-core fiber fallsinto three types or categories: bend-induced strain (B), strain as aresult of torque (R), and axially-applied strain (A). The strainresponse of a core within the fiber possesses a component of theseglobally-applied strains based on its position in the multi-core fiber.The strain response ε of a core at a distance along the fiber can berepresented by equation (1) below:ε_(n) =B _(n)(z)+R _(n)(z)+A _(n)(z)  (1)in which n designates a core within the fiber, z represents an indexalong the fiber length, B is the strain experienced by the core due tobending of the fiber, R is the strain induced in the core by twist ortorsion applied to the fiber, and A represents axial strain experiencedby the core. Compensation for variation in core placement can beachieved by rewriting the expression in equation (1) in terms of theposition of the core using the model parameters established in FIG. 3.The bend strain B perceived by a core as a result of bending of thefiber can be shown to be proportional to curvature of the bend and thetangential distance d of the core to the bend plane (shown in FIG. 3) inEquation (2) below:B _(n)(z)=αK(z)d _(n)(z)  (2)in which α is a constant, K is the curvature of the fiber, and drepresents the tangential distance of the core from the bend plane. Fromthe model in FIG. 3, the tangential distance d can be expressed in termsof the core's position as:d _(n)(z)=r _(n)[sin(ϕ_(n))cos(θ(z))−cos(ϕ_(n))sin(θ(z))]  (3)in which r is the radial distance from the axis of the fiber, ϕrepresents the angle measured from the vertical axis, and θ is a measureof the angle between the bend plane and the horizontal axis. Combiningequations (2) and (3) results in:B _(n)(z)=αK(z)r _(n)[sin(ϕ_(n))cos(θ(z))−cos(ϕ_(n))sin(θ(z))]  (4)This expression can be simplified by distributing the curvature term andexpressing as two separate components:B _(n)(z)=αr _(n)[K _(x)(z)sin(ϕ_(n))−K _(y)(z)cos(ϕ_(n))]  (5)in which K_(x) is the curvature about the horizontal axis (pitch) andK_(y) is the curvature about the vertical axis (yaw).

For moderate levels of twist applied to a fiber (e.g., 100degrees/meter), a first order term can be used to model strain inducedby torque. Twist strain R_(n) (z) is then expressed in terms of the coreposition as follows:R _(n)(z)=βr _(n) ²Φ(z)  (6)in which β is a constant, and Φ is the amount the fiber has twisted(roll), per unit of length. To a first order, it can also be assumedthat the axial strain A experienced by the cores is common to all coreswithin the fiber and is not dependent on the position of the cores toarrive at the expression:A _(n)(z)=γE(z)  (7)in which γ is a constant, and E represents axial strain. Rewritingequation (1) in terms of the core positions results in the followingexpression:ε_(n)(z)=αr _(n) K _(x)(z)sin(ϕ_(n))−αr _(n) K _(y)(z)cos(ϕ_(n))+βr _(n)²Φ(z)+γE(z)  (8)

Considering the measured strain signals from the four cores in thisexample fiber embodiment, a matrix relationship can be constructed asfollows:

$\begin{matrix}{\begin{bmatrix}{ɛ_{0}(z)} \\{ɛ_{1}(z)} \\{ɛ_{2}(z)} \\{ɛ_{3}(z)}\end{bmatrix} = {\begin{bmatrix}{\alpha\; r_{0}{\sin\left( \phi_{0} \right)}} & {{- \alpha}\; r_{0}{\cos\left( \phi_{0} \right)}} & {\beta\; r_{0}^{2}} & \gamma \\{\alpha\; r_{1}{\sin\left( \phi_{1} \right)}} & {{- \alpha}\; r_{1}{\cos\left( \phi_{1} \right)}} & {\beta\; r_{1}^{2}} & \gamma \\{\alpha\; r_{2}{\sin\left( \phi_{2} \right)}} & {{- \alpha}\; r_{2}{\cos\left( \phi_{2} \right)}} & {\beta\; r_{2}^{2}} & \gamma \\{\alpha\; r_{3}{\sin\left( \phi_{3} \right)}} & {{- \alpha}\; r_{3}{\cos\left( \phi_{3} \right)}} & {\beta\; r_{3}^{2}} & \gamma\end{bmatrix}\begin{bmatrix}{K_{x}(z)} \\{K_{y}(z)} \\{\Phi(z)} \\{E(z)}\end{bmatrix}}} & (9)\end{matrix}$

This expression in equation (9) allows recombination of individualstrain signals of each independent core within the shape fiber,according to fiber structure variations, and sorting of these signalsinto strains that are applied to the entire multi-core fiber structure.Any number of linear combinations can be derived from equation (9) tocreate expressions that relate the strain response of a core to acomponent of the strain profile.

If the four parameters K_(x)—the curvature about the horizontal axis(pitch), K_(y)—the curvature about the vertical axis (yaw), Φ—the amountof twist (roll), and E—the axial strain are the only significantdeformations present in the fiber, and the phase deformation in four (4)cores is accurately measured, then the phase in an additional orauxiliary or redundant (5^(th)) core in the fiber may be calculated fromthis four core measurement and compared to a phase measurement for theauxiliary or redundant (5^(th)) core in the fiber. If the phase measuredin the auxiliary (5^(th)) core differs from the phase predicted by theother four cores for the auxiliary (5^(th)) core, then there are twopossibilities to address: the phase deformation measure of at least oneof the five cores is inaccurate, meaning there is an error of some sort,or a physical deformation of the fiber other than the listed fourparameters is present, meaning that the optical shape sensing model orits underlying assumptions is/are incomplete. In either case, thecurrent measurement may be assumed to be flawed, and the shapecalculated may be considered unreliable and a potential hazard.

Consider the following equation that includes the auxiliary or redundant(5^(th)) core in the fiber represented by the variable Δ:

$\begin{matrix}{\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\\Delta\end{bmatrix} = {M\begin{bmatrix}ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4} \\ɛ_{5}\end{bmatrix}}} & (10)\end{matrix}$

A matrix, M, is constructed to calculate the physical parameters and howmuch the measured strain in the 5^(th) core departs from the modeledstrain for Δ.

Equation (9) may be extended to include more cores than just four.Equation (11) shows a five core example.

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix} = {\begin{bmatrix}{\alpha\; r_{0}{\sin\left( \phi_{0} \right)}} & {{- \alpha}\; r_{0}{\cos\left( \phi_{0} \right)}} & {\beta\; r_{0}^{2}} & \gamma_{0} \\{\alpha\; r_{1}{\sin\left( \phi_{1} \right)}} & {{- \alpha}\; r_{1}{\cos\left( \phi_{1} \right)}} & {\beta\; r_{1}^{2}} & \gamma_{1} \\{\alpha\; r_{2}{\sin\left( \phi_{2} \right)}} & {{- \alpha}\; r_{2}{\cos\left( \phi_{2} \right)}} & {\beta\; r_{2}^{2}} & \gamma_{2} \\{\alpha\; r_{3}{\sin\left( \phi_{3} \right)}} & {{- \alpha}\; r_{3}{\cos\left( \phi_{3} \right)}} & {\beta\; r_{3}^{2}} & \gamma_{3} \\{\alpha\; r_{4}{\sin\left( \phi_{4} \right)}} & {{- \alpha}\; r_{4}{\cos\left( \phi_{4} \right)}} & {\beta\; r_{4}^{2}} & \gamma_{4}\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix}}} & (11)\end{matrix}$

The variables are renamed to clean up the notation:

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix} = {\begin{bmatrix}{BY}_{0} & {BX}_{0} & R_{0} & G_{0} \\{BY}_{1} & {BX}_{1} & R_{1} & G_{1} \\{BY}_{2} & {BX}_{2} & R_{2} & G_{2} \\{BY}_{3} & {BX}_{3} & R_{3} & G_{3} \\{BY}_{4} & {BX}_{4} & R_{4} & G_{4}\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix}}} & \left( {11A} \right)\end{matrix}$The matrix in equations (11 and 11A) is not invertible because it is notsquare. Equation 11A is broken into two equations:

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3}\end{bmatrix} = {\begin{bmatrix}{BY}_{0} & {BX}_{0} & R_{0} & G_{0} \\{BY}_{1} & {BX}_{1} & R_{1} & G_{1} \\{BY}_{2} & {BX}_{2} & R_{2} & G_{2} \\{BY}_{3} & {BX}_{3} & R_{3} & G_{3}\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix}}} & \left( {12A} \right)\end{matrix}$which is a reproduction of equation 9, and

$\begin{matrix}{ɛ_{4} = {\left\lbrack {{BY}_{4}\mspace{20mu}{BW}_{4}\mspace{14mu} R_{4}\mspace{20mu} G_{4}} \right\rbrack\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix}}} & \left( {12B} \right)\end{matrix}$

The result is a square matrix in equation (11A) which is invertible andis designated matrix H:

$\begin{matrix}{\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix} = {{\begin{bmatrix}{BY}_{0} & {BX}_{0} & R_{0} & G_{0} \\{BY}_{1} & {BX}_{1} & R_{1} & G_{1} \\{BY}_{2} & {BX}_{2} & R_{2} & G_{2} \\{BY}_{3} & {BX}_{3} & R_{3} & G_{3}\end{bmatrix}^{- 1}\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3}\end{bmatrix}} = {H\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3}\end{bmatrix}}}} & (13)\end{matrix}$

The following substitution is made by substituting the results ofequation 13 into equation 12B:

$\begin{matrix}{ɛ_{4} = {{\left\lbrack {{BY}_{4}\mspace{20mu}{BW}_{4}\mspace{20mu} R_{4}\mspace{20mu} G_{4}} \right\rbrack\begin{bmatrix}{BY}_{0} & {BX}_{0} & R_{0} & G_{0} \\{BY}_{1} & {BX}_{1} & R_{1} & G_{1} \\{BY}_{2} & {BX}_{2} & R_{2} & G_{2} \\{BY}_{3} & {BX}_{3} & R_{3} & G_{3}\end{bmatrix}}^{- 1}\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3}\end{bmatrix}}} & (14)\end{matrix}$which provides an explicit way to calculate a predicted strain in the5^(th) core based on our physical model and the strains measured in theother 4 cores. The above equation may be reduced to an algebraicexpression by defining:

$\begin{matrix}{\left\lbrack {a\mspace{20mu} b\mspace{20mu} c\mspace{25mu} d} \right\rbrack = {\left\lbrack {{BY}_{4}\mspace{20mu}{BW}_{4}\mspace{20mu} R_{4}\mspace{20mu} G_{4}} \right\rbrack\begin{bmatrix}{BY}_{0} & {BX}_{0} & R_{0} & G_{0} \\{BY}_{1} & {BX}_{1} & R_{1} & G_{1} \\{BY}_{2} & {BX}_{2} & R_{2} & G_{2} \\{BY}_{3} & {BX}_{3} & R_{3} & G_{3}\end{bmatrix}}^{- 1}} & (15)\end{matrix}$and writing:ε_(4pred) =aε ₀ +bε ₁ +cε ₂ +dε ₃  (16)

The error is the difference between the predicted strain based on themodel, ε_(4pred), and the actual measured value of strain in the 5^(th)core, ε_(4meas):Δ=ε_(4pred)−ε_(4meas) =aε ₀ +bε ₁ +cε ₂ +dε ₃−ε_(4meas)  (17)

If we further define,

$\begin{matrix}{\begin{bmatrix}h_{00} & h_{01} & h_{02} & h_{03} \\h_{10} & h_{11} & h_{12} & h_{13} \\h_{20} & h_{21} & h_{22} & h_{23} \\h_{30} & h_{31} & h_{32} & h_{33}\end{bmatrix} = \begin{bmatrix}{BY}_{0} & {BX}_{0} & R_{0} & G_{0} \\{BY}_{1} & {BX}_{1} & R_{1} & G_{1} \\{BY}_{2} & {BX}_{2} & R_{2} & G_{2} \\{BY}_{3} & {BX}_{3} & R_{3} & G_{3}\end{bmatrix}^{- 1}} & (18)\end{matrix}$where h_(nm) are the entries of matrix H above in equation 13, then amatrix implementation is constructed for the calculation of the physicalparameter and a measure, Δ, of how the strain in the 5^(th) core differsfrom that predicted by the model.

$\begin{matrix}{\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\\Delta\end{bmatrix} = {\begin{bmatrix}h_{00} & h_{01} & h_{02} & h_{03} & 0 \\h_{10} & h_{11} & h_{12} & h_{13} & 0 \\h_{20} & h_{21} & h_{22} & h_{23} & 0 \\h_{30} & h_{31} & h_{32} & h_{33} & 0 \\a & b & c & d & {- 1}\end{bmatrix}\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix}}} & (19)\end{matrix}$where the matrix we were originally looking for, M, is given by:

$\begin{matrix}{M = \begin{bmatrix}h_{00} & h_{01} & h_{02} & h_{03} & 0 \\h_{10} & h_{11} & h_{12} & h_{13} & 0 \\h_{20} & h_{21} & h_{22} & h_{23} & 0 \\h_{30} & h_{31} & h_{32} & h_{33} & 0 \\a & b & c & d & {- 1}\end{bmatrix}} & (20)\end{matrix}$where h_(nm) are the entries of matrix H above in equation 13. The Hmatrix relates a set of strains to an equal number of physicalparameters, while the M matrix includes a calculated error parameter.

In a practical example embodiment, to find the strain (E) and error (Δ)independently, the other three parameters (twist (Φ), bend-x (K_(x)),and bend-y (K_(y))) are determined, and the fiber is calibrated for allof these effects.

The calibration begins by determining the core geometries (radii andangles) for all five cores, (see FIG. 3). By measuring the strain in thecores at different tensions, values for the parameters γ_(n) forequation (11) are determined. From these data sets, the matrix inequation (11) is determined for calculating fiber pitch (K_(x)), yaw(K_(y)), twist (Φ), strain (E), and error (Δ) from the OFDR measurementsfor the five cores.

$\begin{matrix}{\begin{bmatrix}{pitch} \\{yaw} \\{twist} \\{strain} \\{error}\end{bmatrix} = {\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\\Delta\end{bmatrix} = {{M\begin{bmatrix}ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4} \\ɛ_{5}\end{bmatrix}} =}}} & (21)\end{matrix}$

Keeping in mind the matrix relating the four low-force stimuli(temperature is not considered here) to the strains ε₀-ε₄ in the fivecores defined in equation (11), the fiber is placed in a continuous bendin a plane. Once the fiber is placed in this configuration, a multiplechannel OFDR system measures the distributed strain response of each ofthe cores within the multi-core optical fiber. The resulting strainresponse signal typically alternates between compression and elongationat a frequency that matches the spin frequency of the helical wrappingof an outer core as it proceeds through a bend. The magnitude of thisoscillation should also be slowly varying along the length of the fiberas this magnitude will be proportional the bend radius of the loopdescribed above. These strain responses are then provided to one or moredata processors for extraction of the parameters that quantify variationfrom an ideal fiber structure.

To determine the core location, a complex-valued signal with bothamplitude and phase is determined from the real-valued strain responseprovided by the OFDR system. A Fourier transform allows a filter to beapplied to the measured scatter signal at the spin frequency of thehelical wrapping. An inverse Fourier transform of this filtered signalproduces a complex-valued spin signal. The amplitude of this complexspin signal is proportional to the radial separation distance of thecore from the neutral center axis of the fiber. The phase response ofthe complex spin signal is based on the angular position of the corewithin the geometry of the fiber. Comparing the complex spin signal of acore to the spin signal of a reference core determines that core'sposition relative to the reference core. Thus, all angular positions canbe found relative to the vertical axis that bisects the reference coreby extracting the argument of a complex quotient between a core's spinsignal and the reference core spin signal. Extracting the amplitudeprovides a ratio measurement of radial separation of the core relativeto the reference core.

From this, the matrix below is populated, where the magnitude of thebend response is still unknown because the bend amplitude of the fiberin the spiral in-plane configuration (X and Y) is still not known.

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix} = {\begin{bmatrix}{\alpha\; Y_{0}} & {{- \alpha}\; X_{0}} & 0 & 0 \\{\alpha\; Y_{1}} & {{- \alpha}\; X_{1}} & 0 & 0 \\{\alpha\; Y_{2}} & {{- \alpha}\; X_{2}} & 0 & 0 \\{\alpha\; Y_{3}} & {{- \alpha}\; X_{3}} & 0 & 0 \\{\alpha\; Y_{4}} & {{- \alpha}\; X_{3}} & 0 & 0\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix}}} & (22)\end{matrix}$A twist is applied to the fiber without changing its shape. From thisthe response of each core to twist alone (R) may be determined.

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix} = {\begin{bmatrix}{\alpha\; Y_{0}} & {{- \alpha}\; X_{0}} & R_{0} & 0 \\{\alpha\; Y_{1}} & {{- \alpha}\; X_{1}} & R_{1} & 0 \\{\alpha\; Y_{2}} & {{- \alpha}\; X_{2}} & R_{2} & 0 \\{\alpha\; Y_{3}} & {{- \alpha}\; X_{3}} & R_{3} & 0 \\{\alpha\; Y_{4}} & {{- \alpha}\; X_{3}} & R_{4} & 0\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix}}} & (23)\end{matrix}$Putting the fiber into a known bend provides known amplitudes for thebend coefficients (B).

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix}=={\begin{bmatrix}{BY}_{0} & {- {BX}_{0}} & R_{0} & 0 \\{BY}_{1} & {- {BX}_{1}} & R_{1} & 0 \\{BY}_{2} & {- {BX}_{2}} & R_{2} & 0 \\{BY}_{3} & {- {BX}_{3}} & R_{3} & 0 \\{BY}_{4} & {- {BX}_{3}} & R_{4} & 0\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix}}} & (24)\end{matrix}$Putting the fiber in a straight line tension (G) allows a determinationof the response of each core to the tension (G).

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix}=={\begin{bmatrix}{BY}_{0} & {- {BX}_{0}} & R_{0} & G_{0} \\{BY}_{1} & {- {BX}_{1}} & R_{1} & G_{1} \\{BY}_{2} & {- {BX}_{2}} & R_{2} & G_{2} \\{BY}_{3} & {- {BX}_{3}} & R_{3} & G_{3} \\{BY}_{4} & {- {BX}_{3}} & R_{4} & G_{4}\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E\end{bmatrix}}} & (25)\end{matrix}$Using this matrix, we can then find the matrix M as described above.

FIG. 4 shows a schematic diagram of a first example embodiment of anoptical frequency domain reflectometry (OFDR)-based shape sensing systemthat identifies errors independent from the model and assumptions of theshape sensing system using a five core fiber.

An OFDR-based distributed strain sensing system includes a light source11, an interferometric interrogator 15, a laser monitor network 12, anoptical sensing fiber 17 that is a multicore shape sensing fiber,acquisition electronics 18, and a data processor 20. A single channelcorresponds to a single fiber core. During an OFDR measurement, atunable light source 11 is swept through a range of optical frequencies.This light is split with the use of optical couplers and routed toseparate interferometers. A laser monitor network 12 contains a HydrogenCyanide (HCN) gas cell that provides an absolute wavelength referencethroughout the measurement scan. An interferometer within a lasermonitor network 12 is used to measure fluctuations in tuning rate as thelight source is scanned through a frequency range.

Interferometric interrogators 15 are connected to respective individualcores in a length of sensing fiber 17 that is a multicore shape sensingfiber. Light enters the sensing fiber 17 through the measurement arms ofthe five interferometric interrogators referenced generally at 15corresponding to five core waveguides A, B, C, D, and E in the fiber 17.Scattered light from each core in the sensing fiber 17 is theninterfered with light that has traveled along the reference arm of thecorresponding interferometric interrogator 15. Although the term core isused, the technology applies to other types of waveguides that can beused in a spun fiber. Each pairing of an interferometric interrogatorwith a waveguide in the multi-core fiber is referred to as anacquisition channel. As the tunable light source 11 is swept, eachchannel is simultaneously measured, and the resulting interferencepattern from each channel is routed to the data acquisition electronics18 adapted for the additional interferometric interrogators 15. Eachchannel is processed independently and identically.

A series of optical detectors (e.g., photodiodes) convert the lightsignals from the laser monitor network, gas cell, and the interferencepatterns from each core from the sensing fiber to electrical signals.One or more data processors in data acquisition electronics 18 uses theinformation from the laser monitor network 12 to resample the detectedinterference patterns of the sensing fiber 16 so that the patternspossess increments constant in optical frequency. This step is amathematical requisite of the Fourier transform operation. Onceresampled, a Fourier transform is performed by the system controllerdata processor 20 to produce a light scatter signal in the temporaldomain. In the temporal domain, the amplitudes of the light scatteringevents are depicted verses delay along the length of the fiber. Usingthe distance that light travels in a given increment of time, this delaycan be converted to a measure of length along the sensing fiber. Inother words, the light scatter signal indicates each scattering event asa function of distance along the fiber. The sampling period is referredto as the spatial resolution and is inversely proportional to thefrequency range that the tunable light source 10 was swept throughduring the measurement.

As the fiber is strained, the local light scatters shift as the fiberchanges in physical length. These distortions are highly repeatable.Hence, an OFDR measurement of detected light scatter for the fiber canbe retained in memory that serves as a reference pattern of the fiber inan unstrained state. A subsequently measured scatter signal when thefiber is under strain may then be compared to this reference pattern bythe system controller data processor 20 to gain a measure of shift indelay of the local scatters along the length of the sensing fiber. Thisshift in delay manifests as a continuous, slowly varying optical phasesignal when compared against the reference scatter pattern. Thederivative of this optical phase signal is directly proportional tochange in physical length of the sensing core. Change in physical lengthmay be scaled to strain thereby producing a continuous measurement ofstrain along the sensing fiber.

The data processor 22 coupled to the system controller data processor 20extracts parameters 24 relating to the actual physical configuration ofthe cores a, b, c, d, and e in fiber 17 that are used to calibrate orotherwise compensate the OFDR measurements to account for the variationsbetween the actual optical core configuration and an optimal opticalcore configuration. The mathematical model described in detail above isfirst established that depicts parameters that describe variations froman optimal multi-core fiber configuration, where the term “optimal”includes known and unknown configurations. Parameters are then definedthat compensate for variation in the physical properties of the opticalcores within the multi-core fiber.

FIG. 5 is a flowchart diagram for calibrating a five core optical shapesensing fiber. Initially, the multicore fiber is placed in a straightline, unstrained configuration, an OFDR measurement is performed (stepS1) as described above, and the resulting reference state parameters arestored (step S2). The multicore fiber is then configured in a knownconfiguration such as in a flat plane, in a helical shape (e.g., ascrew), etc. In a non-limiting example, the multicore fiber isconfigured in a flat plane (step S3) to calculate the relative geometrybetween the cores in the fiber (step S4). A twist is applied in thisconfiguration (step S5), and a twist response is determined (step S6).The fiber is then configured into a known bend position (step S7), and abend gain is calculated that provides amplitude values of the coregeometry (step S8). The fiber is placed under tension (step S9), and atension response for each core calculated (step S10). The values neededto populate the matrix in equation (25) above, which describes theresponse of the five cores to bend, strain, and twist, are thenavailable (step S11), and the matrix can be calculated using the stepsdescribed above in equations (15)-(20). The error term, Δ may then beused to detect one or more errors in shape sensing applications usingthat shape sensing fiber such as pinch, errors in the electronics, etc.

FIG. 6 is a flowchart diagram carried out by the system controller fordetecting an error in shape sensing system using a calibrated shapesensing fiber having four primary cores and one auxiliary core inaccordance with the first example embodiment. Initially, the calibratedshape sensing fiber is placed as desired for shape sensing, and OFDRscatter measurements are obtained for each of the five cores (fourprimary cores and one secondary core (also called auxiliary core orredundant core)) (step S20). The data processor 22 tracks the opticalphase signal for each core determined from these scatter measurements ascompared to the calibrated reference scatter patterns for eachcorresponding core for this fiber (step S21). Each of the optical phasesignals is a measure of shift in delay of the local scatters along thelength of its respective core in the sensing fiber. The derivative ofthis optical phase signal is calculated for each of the four primarycores (step S22), which is directly proportional to change in physicallength of its respective core. Each of the four phase derivatives ismultiplied by the conversion matrix M from equation (20) to determinethe applied bend, strain, and twist, and then the parameters describingthe 5^(th) are used to produce a measurement of the predicted phasederivative of the auxiliary core (step S23) and the measured phasederivative of the auxiliary core.

If the difference between the predicted measurement of the auxiliary andactual measurement of the auxiliary core differ by more than apredetermined amount, (one non-limiting example amount might be 0.5radians), the OFDR shape sensing measurements are labeled unreliableand/or one or more the following actions is taken or initiated: generatea fault signal for display, stop operation of the system or machineassociated with the shape sensing fiber, generate an alarm, and/or takesome other precautionary or protection action (step S26).

Second Example Embodiment

Temperature can also change the apparent length of the cores andrepresents a fifth degree of “freedom” in the system. Many shape sensingsystems do not distinguish between temperature changes along the lengthof the fiber and axial strain changes along the length of the fiber.

A shape sensing system is now described that mitigates or compensatesfor the errors imposed by differences in a shape sensing optical fiber'sresponse to temperature and strain. In other words, the shape sensingmodel in the second example embodiment explicitly addresses andcompensates for the effect of temperature in the shape sensingcalculations. In contrast, the first example embodiment did not, whichmeans that any temperature effects on the shape sensing determinationsin the first embodiment are detected as part of the error calculationdescribed above.

However, in this second example embodiment, a fifth core is incorporatedas one of five primary cores that has a different temperature dependencefrom the other four primary cores in a twisted multicore optical fiber.In addition, a sixth auxiliary core is added. It should be appreciatedthat while much of the description below is in the context of these fiveand six core examples, the principles described in the first and secondembodiments apply to twisted multicore fibers with different numbers ofcores. The temperature sensing or auxiliary cores are preferably locatedin the fiber in such a way so as to reduce or minimize coupling betweencores.

In one example embodiment, the fifth primary core has a differentthermal dependence by having a different refractive index from that ofthe other four primary cores. Example ways to achieve that differentrefractive index include the fifth primary core being composed of adifferent material and/or being differently doped. Other ways arepossible to achieve a different temperature dependence such as forexample locating a fifth primary core at a different radius (closer orfurther from the center of the fiber) than the other four primary cores,providing the fifth primary core with a different geometry (bigger orsmaller than the other cores), etc.

Four of the primary cores may be doped with germanium, and a fifthprimary core may be co-doped with boron and germanium. Boron dopinginduces thermal stress in polarization maintaining optical fiber, and asa result, the thermal response of a core containing boron to has adifferent thermal dependence as compared to typical germanium-dopedfiber. Although other dopants may be used to create a different thermaldependence as compared to typically doped fibers in a multicore fiber,boron is also used in example embodiments as a co-dopant with germaniumbecause that co-doping also forms a more photo-sensitive guiding core ascompared to typical germanium-doped fiber. Moreover, boron is arelatively common dopant for optical fiber which provides furtherpractical advantages.

FIG. 7 shows a graph of index of refraction versus temperature for aboron-geranium co-doped core and a geranium doped core. As temperatureincreases, the index of refraction for a boron-germanium co-doped coreincreases at a different (higher) rate than the index of refraction fora typical germanium-doped core. The fifth primary core reactsdifferently to temperature changes, and therefore, provides anadditional, linearly-independent source of information that is used tocompensate for temperature.

Errors and/or uncertainties arise in the determination of fiber positionand/or shape—and more generally strain—due to variations in thestructure of the multi-core optical fiber. The first category ofvariation is core placement. This variation causes both the radialseparation and the angular position of a given core to differ fromdesigned or desired ideal values or to simply be unknown. A mathematicalmodel is generated that describes the positions of the cores withrespect to the cross section of the multi-core fiber such thatvariations can be quantified.

As glass is a relatively hard material, it can be assumed that thegeometry of the cross section of the multi-core fiber is preserved asthe fiber is strained. This assures that the relative positions of thecores within a given cross section remain constant as the fiber isstrained. This means that the fiber can be strained and still be used toaccurately determine variations in core placement from the idealconfiguration. A core position model that accounts for variation in coreplacement for a fiber with six cores (a-f) is depicted in FIG. 8. Theauxiliary or redundant core is labeled fin FIG. 8, and the differencebetween the radius distances for peripheral cores b-e and the radiusdistance for peripheral core f is more than an insubstantial difference.For example, the difference divided by the average radius distance forall of peripheral cores b-f is 0.10 or more. Stated differently, thedifference is 10% of the average radius distance for all of peripheralcores b-f. As in FIG. 3, the auxiliary or redundant core f mayalternatively have a substantially longer radius than peripheral coresb-e, where again the radial distance difference is 10% of the averageradius distance for all of cores b-f. In some embodiments, the averageradius distance is calculated as the mean radius distance.

In an alternative example embodiment, the auxiliary redundant core f islocated at the same radius as the other peripheral cores. Although theauxiliary redundant core may still be used to check the reliability ofthe data, this configuration is less effective at detecting errors onthe central core.

In line with the equations already presented above in the firstembodiment, if temperature is allowed to be an independent variable,then adding a 6^(th) core changes our starting equation from this:

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix} = {\begin{bmatrix}{BY}_{0} & {- {BX}_{0}} & R_{0} & G_{0} & T_{0} \\{BY}_{1} & {- {BX}_{1}} & R_{1} & G_{1} & T_{1} \\{BY}_{2} & {- {BX}_{2}} & R_{2} & G_{2} & T_{2} \\{BY}_{3} & {- {BX}_{3}} & R_{3} & G_{3} & T_{3} \\{BY}_{4} & {- {BX}_{4}} & R_{4} & G_{4} & T_{4}\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\T\end{bmatrix}}} & (26)\end{matrix}$to this:

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4} \\ɛ_{5}\end{bmatrix} = {\begin{bmatrix}{BY}_{0} & {BX}_{0} & R_{0} & G_{0} & T_{0} \\{BY}_{1} & {BX}_{1} & R_{1} & G_{1} & T_{1} \\{BY}_{2} & {BX}_{2} & R_{2} & G_{2} & T_{2} \\{BY}_{3} & {BX}_{3} & R_{3} & G_{3} & T_{3} \\{BY}_{4} & {BX}_{4} & R_{4} & G_{4} & T_{4} \\{BY}_{5} & {BX}_{5} & R_{5} & G_{5} & T_{5}\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\T\end{bmatrix}}} & (27)\end{matrix}$Pulling out the last row produces an expression for the strain in the6^(th) core, ε_(5pred), as predicted by the physical state of the fiber(bend, twist, strain and temperature):

$\begin{matrix}{ɛ_{5} = {\begin{bmatrix}{BY}_{5} & {BX}_{5} & R_{5} & G_{5} & T_{5}\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\T\end{bmatrix}}} & (28)\end{matrix}$All of these physical parameters can be determined from the other fivecores:

$\begin{matrix}{{\begin{bmatrix}{BY}_{0} & {- {BX}_{0}} & R_{0} & G_{0} & T_{0} \\{BY}_{1} & {- {BX}_{1}} & R_{1} & G_{1} & T_{1} \\{BY}_{2} & {- {BX}_{2}} & R_{2} & G_{2} & T_{2} \\{BY}_{3} & {- {BX}_{3}} & R_{3} & G_{3} & T_{3} \\{BY}_{4} & {- {BX}_{3}} & R_{4} & G_{4} & T_{4}\end{bmatrix}^{- 1}\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix}} = {\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\T\end{bmatrix}.}} & (29)\end{matrix}$Substitution is used to obtain:

$\begin{matrix}{ɛ_{5\;{pred}} = {{\begin{bmatrix}{BY}_{5} & {BX}_{5} & R_{5} & G_{5} & T_{5}\end{bmatrix}\begin{bmatrix}{BY}_{0} & {- {BX}_{0}} & R_{0} & G_{0} & T_{0} \\{BY}_{1} & {- {BX}_{1}} & R_{1} & G_{1} & T_{1} \\{BY}_{2} & {- {BX}_{2}} & R_{2} & G_{2} & T_{2} \\{BY}_{3} & {- {BX}_{3}} & R_{3} & G_{3} & T_{3} \\{BY}_{4} & {- {BX}_{3}} & R_{4} & G_{4} & T_{4}\end{bmatrix}}^{- 1}\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4}\end{bmatrix}}} & (30)\end{matrix}$which is a closed form solution for the predicted value of the strain inthe 6^(th) core. This can be reduced to a row operation:

$\begin{matrix}{\begin{bmatrix}a & b & c & d & e\end{bmatrix} = {\begin{bmatrix}{BY}_{5} & {BX}_{5} & R_{5} & G_{5} & T_{5}\end{bmatrix}\begin{bmatrix}{BY}_{0} & {- {BX}_{0}} & R_{0} & G_{0} & T_{0} \\{BY}_{1} & {- {BX}_{1}} & R_{1} & G_{1} & T_{1} \\{BY}_{2} & {- {BX}_{2}} & R_{2} & G_{2} & T_{2} \\{BY}_{3} & {- {BX}_{3}} & R_{3} & G_{3} & T_{3} \\{BY}_{4} & {- {BX}_{3}} & R_{4} & G_{4} & T_{4}\end{bmatrix}}^{- 1}} & (31)\end{matrix}$to produce an algebraic expression for the predicted strain on the6^(th) core:ε_(5pred) =aε ₀ +bε ₁ +cε ₂ +dε ₃ +eε ₄  (32)Subtracting the measured strain ε_(5meas) from the predicted strainprovides the error term, Δ:ε_(5pred)−ε_(5meas)=ε_(error) =Δ=aε ₀ +bε ₁ +cε ₂ +dε ₃ +eε₄−ε_(5meas)  (33)The inversion matrix of the physical parameters is calculated andexpressed as a matrix of entries:

$\begin{matrix}{\begin{bmatrix}{BY}_{0} & {- {BX}_{0}} & R_{0} & G_{0} & T_{0} \\{BY}_{1} & {- {BX}_{1}} & R_{1} & G_{1} & T_{1} \\{BY}_{2} & {- {BX}_{2}} & R_{2} & G_{2} & T_{2} \\{BY}_{3} & {- {BX}_{3}} & R_{3} & G_{3} & T_{3} \\{BY}_{4} & {- {BX}_{3}} & R_{4} & G_{4} & T_{4}\end{bmatrix}^{- 1} = {\begin{bmatrix}h_{00} & h_{01} & h_{02} & h_{03} & h_{04} \\h_{10} & h_{11} & h_{12} & h_{13} & h_{14} \\h_{20} & h_{21} & h_{22} & h_{23} & h_{24} \\h_{30} & h_{31} & h_{32} & h_{33} & h_{34} \\h_{40} & h_{41} & h_{42} & h_{43} & h_{44}\end{bmatrix} = H}} & (34)\end{matrix}$A concise expression is constructed for the calculation of all of thephysical parameters as well as the new parameter Δ that is a measure ofthe how well the measured strains match the model:

$\begin{matrix}{{M\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4} \\ɛ_{5}\end{bmatrix}} = {{\begin{bmatrix}h_{00} & h_{01} & h_{02} & h_{03} & h_{04} & 0 \\h_{10} & h_{11} & h_{12} & h_{13} & h_{14} & 0 \\h_{20} & h_{21} & h_{22} & h_{23} & h_{24} & 0 \\h_{30} & h_{31} & h_{32} & h_{33} & h_{34} & 0 \\h_{40} & h_{41} & h_{42} & h_{43} & h_{44} & 0 \\a & b & c & d & e & {- 1}\end{bmatrix}\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4} \\ɛ_{5}\end{bmatrix}} = \begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\T \\\Delta\end{bmatrix}}} & (35)\end{matrix}$

In a practical example for the second embodiment, to find the strain (E)and temperature (T) independently, the other three parameters (twist(Φ), bend-x (K_(x)), and bend-y (K_(y))) are determined, and the fiberis calibrated for all of these effects.

The calibration begins by determining the core geometries (radii andangles) for all five cores, (see FIG. 8), and then suspending the fiberinside a tube furnace or other suitable temperature-controlledenvironment. By measuring the strain in the cores at differenttemperatures and different strains, values for the parameters G_(n) andT_(n) for equation (27) are determined. The six core optical sensingfiber is calibrated with an additional set of data taken with the fiberunder tension, and one more set of data with the fiber heated. Fromthese data sets, the matrix in equation (29) is determined forcalculating fiber pitch (K_(x)), yaw (K_(y)), twist (Φ), strain (E), andtemperature (T) from the OFDR measurements for the five cores.

$\begin{matrix}{\begin{bmatrix}{pitch} \\{yaw} \\{twist} \\{strain} \\{temperature} \\\Delta\end{bmatrix} = {\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\T \\\Delta\end{bmatrix} = {\overset{\_}{\overset{\_}{M}}\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4} \\ɛ_{5}\end{bmatrix}}}} & (36)\end{matrix}$

The calibration procedure and equations are similar to those used in thefirst embodiment with the additional calibration for temperature.Elevating the temperature of the fiber allows determination of thetemperature response (T) of each core.

$\begin{matrix}{\begin{bmatrix}ɛ_{0} \\ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4} \\ɛ_{5}\end{bmatrix} = {\begin{bmatrix}{BY}_{0} & {- {BX}_{0}} & R_{0} & G_{0} & T_{0} \\{BY}_{1} & {- {BX}_{1}} & R_{1} & G_{1} & T_{1} \\{BY}_{2} & {- {BX}_{2}} & R_{2} & G_{2} & T_{2} \\{BY}_{3} & {- {BX}_{3}} & R_{3} & G_{3} & T_{3} \\{BY}_{4} & {- {BX}_{4}} & R_{4} & G_{4} & T_{4} \\{BY}_{5} & {- {BX}_{5}} & R_{5} & G_{5} & T_{5}\end{bmatrix}\begin{bmatrix}K_{x} \\K_{y} \\\Phi \\E \\T\end{bmatrix}}} & (37)\end{matrix}$

This completes the matrix relating the individual core responses to theaggregate linear effects of Pitch, Yaw, Twist, Tension, and Temperature.This matrix is used to calculate the error term and all of the physicalparameters as described above.

FIG. 9 shows a schematic diagram of a second example embodiment of anOFDM-based shape sensing system that compensates for temperature andidentifies errors independent from the model and assumptions of theshape sensing system using a six core fiber. FIG. 9 is similar to FIG. 4with the addition of optics and processing to accommodate six cores(a-f) in the fiber 17.

Interferometric interrogators 15 are connected to respective individualcores in a length of sensing fiber 17. Light enters the sensing fiber 17through the measurement arms of the five interferometric interrogatorsreferenced generally at 15 corresponding to six core waveguides a, b, c,d, e, and fin the fiber 17. Scattered light from each core the sensingfiber 17 is then interfered with light that has traveled along thereference arm of the corresponding interferometric interrogator 15. Asthe tunable light source 10 is swept, each channel is simultaneouslymeasured, and the resulting interference pattern from each channel isrouted to the data acquisition electronics 18 adapted for the additionalinterferometric interrogators 15. Each channel is processedindependently and identically using the OFDR procedures described abovebut also processing the temperature sensing and error detection cores.

FIG. 10 is a flowchart diagram for calibrating a six core optical shapesensing fiber. Steps S1-S11 are the same as for FIG. 5. In addition, thefiber is exposed to increased temperature in a temperature controlledenvironment (step S12), and a temperature response is calculated foreach core (step S13). The values needed to populate the matrix inequation (37) described above are then available (step S11), and thatmatrix M can be calculated (equations (31)-(35)) and used to compensatefor temperature in shape sensing applications using that shape sensingfiber and to detect any error.

FIG. 11 is a flowchart diagram for detecting an error in shape sensingsystem of the second example embodiment. Initially, the calibrated shapesensing fiber is placed as desired for shape sensing, and OFDR scattermeasurements are obtained for each of the six cores (step S30). The dataprocessor 22 tracks the optical phase signal for each core determinedfrom these scatter measurements as compared to the calibrated referencescatter patterns for each corresponding core for this fiber (step S31).Each of the optical phase signals is a measure of shift in delay of thelocal scatters along the length of its respective core in the sensingfiber. The derivative of this optical phase signal is calculated foreach of the five primary cores (step S32), which is directlyproportional to change in physical length of its respective core. Eachof the six phase derivatives is multiplied by the matrix M to calculatethe applied bend, twist, strain, and an error term. If this error termexceeds a certain magnitude, the OFDR shape sensing measurements arelabeled unreliable and/or one or more the following actions is taken orinitiated: generate a fault signal for display, stop operation of thesystem or machine associated with the shape sensing fiber, generate analarm, and/or take some other precautionary or protection action (stepS36).

If new sensed parameters are added in the same way that temperaturesensing was added in the second example embodiment, then an extra coreis for redundancy. The addition of more than one redundant core isreadily accommodated using the approach described above and providesfurther assurance of the reliability and integrity of shape sensingmeasurements.

Whenever it is described in this document that a given item is presentin “some embodiments,” “various embodiments,” “certain embodiments,”“certain example embodiments, “some example embodiments,” “an exemplaryembodiment,” or whenever any other similar language is used, it shouldbe understood that the given item is present in at least one embodiment,though is not necessarily present in all embodiments. Consistent withthe foregoing, whenever it is described in this document that an action“may,” “can,” or “could” be performed, that a feature, element, orcomponent “may,” “can,” or “could” be included in or is applicable to agiven context, that a given item “may,” “can,” or “could” possess agiven attribute, or whenever any similar phrase involving the term“may,” “can,” or “could” is used, it should be understood that the givenaction, feature, element, component, attribute, etc. is present in atleast one embodiment, though is not necessarily present in allembodiments. Terms and phrases used in this document, and variationsthereof, unless otherwise expressly stated, should be construed asopen-ended rather than limiting. As examples of the foregoing: “and/or”includes any and all combinations of one or more of the associatedlisted items (e.g., a and/or b means a, b, or a and b); the singularforms “a”, “an” and “the” should be read as meaning “at least one,” “oneor more,” or the like; the term “example” is used provide examples ofthe subject under discussion, not an exhaustive or limiting listthereof; the terms “comprise” and “include” (and other conjugations andother variations thereof) specify the presence of the associated listeditems but do not preclude the presence or addition of one or more otheritems; and if an item is described as “optional,” such descriptionshould not be understood to indicate that other items are also notoptional.

As used herein, the term “non-transitory computer-readable storagemedium” includes a register, a cache memory, a ROM, a semiconductormemory device (such as a D-RAM, S-RAM, or other RAM), a magnetic mediumsuch as a flash memory, a hard disk, a magneto-optical medium, anoptical medium such as a CD-ROM, a DVD, or Blu-Ray Disc, or other typeof device for non-transitory electronic data storage. The term“non-transitory computer-readable storage medium” does not include atransitory, propagating electromagnetic signal.

Although various embodiments have been shown and described in detail,the claims are not limited to any particular embodiment or example. Thetechnology fully encompasses other embodiments which may become apparentto those skilled in the art. None of the above description should beread as implying that any particular element, step, range, or functionis essential such that it must be included in the claims scope. Thescope of patented subject matter is defined only by the claims. Theextent of legal protection is defined by the words recited in the claimsand their equivalents. All structural and functional equivalents to theelements of the above-described preferred embodiment that are known tothose of ordinary skill in the art are expressly incorporated herein byreference and are intended to be encompassed by the present claims.Moreover, it is not necessary for a device or method to address each andevery problem sought to be solved by the technology described, for it tobe encompassed by the present claims. No claim is intended to invokeparagraph 6 of 35 USC § 112 unless the words “means for” or “step for”are used. Furthermore, no embodiment, feature, component, or step inthis specification is intended to be dedicated to the public regardlessof whether the embodiment, feature, component, or step is recited in theclaims.

What is claimed is:
 1. An interferometric measurement system formeasuring an optical fiber including multiple primary cores configuredin the optical fiber and an auxiliary core configured in the opticalfiber, the system comprising: interferometric detection circuitryconfigured to detect measurement interferometric pattern data associatedwith each of the multiple primary cores and the auxiliary core; and dataprocessing circuitry configured to: determine compensation parametersbased on the detected measurement interferometric pattern data for themultiple primary cores, the compensation parameters compensating forvariations between a calibration configuration of the multiple primarycores and an actual configuration of the multiple primary cores, comparea predicted parameter value for the auxiliary core with ameasurement-based parameter value for the auxiliary core to produce acomparison, determine an unreliability of the determined compensationparameters based on the comparison, and generate a signal in response tothe unreliability.
 2. The interferometric measurement system in claim 1,wherein the signal represents an error comprising: (a) an error inoperation of the detection circuitry or the data processing circuitry,(b) an error in an optical fiber connection, (c) an error in thecalibration configuration, or (d) an error caused by a force experiencedby the optical fiber for which a compensation parameter is notdetermined by the data processing circuitry.
 3. The interferometricmeasurement system in claim 2, wherein the error is caused by a pinchingof the optical fiber.
 4. The interferometric measurement system claim 2,wherein the error is caused by a change in temperature.
 5. Theinterferometric measurement system in claim 1, wherein the dataprocessing circuitry is further configured to apply the compensationparameters to subsequently-obtained measurement interferometric patterndata for the optical fiber.
 6. The interferometric measurement system inclaim 1, wherein the predicted parameter value is a predicted phase forthe auxiliary core and the measurement-based parameter value is ameasurement-based phase for the auxiliary core.
 7. The interferometricmeasurement system in claim 6, wherein the data processing circuitry isfurther configured to determine the predicted phase for the auxiliarycore by: calculating a derivative of a phase measured in each of themultiple primary cores to obtain multiple phase derivatives; multiplyingthe multiple phase derivatives by a conversion matrix to obtain apredicted auxiliary core phase derivative; and integrating the predictedauxiliary core phase derivative to obtain the predicted phase for theauxiliary core.
 8. The interferometric measurement system in claim 1,wherein the data processing circuitry is further configured to:determine strain values for the optical fiber based on the detectedmeasurement interferometric pattern data, the strain valuescorresponding to an axial strain, a bend strain, and a twist strain onthe optical fiber, and determine a shape of the optical fiber based onthe strain values for the optical fiber.
 9. The interferometricmeasurement system in claim 1, wherein the data processing circuitry isconfigured to: determine strain values for the optical fiber based onthe detected measurement interferometric pattern data; the strain valuescorresponding to an axial strain, a bend strain, a twist strain, and atemperature strain on the optical fiber, and determine a shape of theoptical fiber based on the strain values for the optical fiber.
 10. Theinterferometric measurement system in claim 1, wherein the dataprocessing circuitry is further configured to generate the signal whenthe unreliability exceeds a predetermined threshold.
 11. Theinterferometric measurement system in claim 1, wherein the signal isrepresentative of the unreliability.
 12. The interferometric measurementsystem in claim 1, wherein the interferometric detection circuitry isconfigured to detect the measurement interferometric pattern data whenthe optical fiber is placed into a sensing position.
 13. Aninterferometric measurement method for measuring an optical fiberincluding multiple primary cores configured in the optical fiber and anauxiliary core configured in the optical fiber, the method comprising:detecting, using interferometric detection circuitry, measurementinterferometric pattern data associated with each of the multipleprimary cores and the auxiliary core when the optical fiber is in asensing position; and determining, using data processing circuitry,compensation parameters based on the detected measurementinterferometric pattern data for the multiple primary cores, thecompensation parameters compensating for variations between acalibration configuration of the multiple primary cores and an actualconfiguration of the multiple primary cores, comparing a predictedparameter value for the auxiliary core with a measurement-basedparameter value for the auxiliary core to produce a comparison,determining an unreliability of the compensation parameters based on thecomparison, and generating a signal in response to the unreliability.14. The interferometric measurement method in claim 13, wherein thesignal represents an error comprising: (a) an error in operation of thedetection circuitry or the data processing circuitry, (b) an error in anoptical fiber connection, (c) an error in the calibration configuration,or (d) an error caused by a force experienced by the optical fiber forwhich a compensation parameter is not determined by the data processingcircuitry.
 15. The interferometric measurement method in claim 13,further comprising applying the compensation parameters tosubsequently-obtained measurement interferometric pattern data for theoptical fiber.
 16. The interferometric measurement method in claim 13,further comprising: determining strain values for the optical fiberbased on the detected measurement interferometric pattern data, thestrain values corresponding to an axial strain, a bend strain, and atwist strain on the optical fiber, and determining a shape of theoptical fiber based on the strain values.
 17. The interferometricmeasurement method in claim 13, wherein generating the signal inresponse to the unreliability comprises: generating the signal when theunreliability exceeds a predetermined threshold.
 18. An optical fibercomprising: a plurality of cores comprising: a central core placed alonga neutral axis of the fiber, and a plurality of peripheral cores, theplurality of peripheral cores comprising four or more primary peripheralcores and an auxiliary peripheral core, wherein each peripheral core ofthe plurality of peripheral cores is at a respective radius distancefrom the central core, wherein at least one of the four or more primaryperipheral cores includes a different dopant than the remaining primaryperipheral cores to provide a temperature response different than atemperature response of the remaining primary peripheral cores of theplurality of cores, and wherein the four or more primary peripheralcores are at a first radius distance from the central core and theauxiliary peripheral core is at a second radius distance from thecentral core, and wherein the plurality of peripheral cores has anaverage radius distance from the central core, and wherein a differencebetween the first radius distance and the second radius distance is atleast 10% of the average radius distance.
 19. The optical fiber in claim18, wherein the plurality of peripheral cores is helically-twisted alonga length of the optical fiber.
 20. The optical fiber of claim 18,wherein the at least one of the four or more primary peripheral cores isboron-germanium co-doped and the remaining primary peripheral cores aredoped with germanium and not boron.